# Full text of "Analytical Mechanics"

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```EQUILIBRIUM OF RIGID BODIES                  39
venient to disregard the reaction of the axle. When this is done the torque of the couple is called the moment of the force applied. Therefore the moment of a force about an axis equals the product of the force by its lever-arm. The lever-arm of a force is the shortest distance between the axis and the line of action of the force. In Fig. 31 the moment of F about the axis through the point 0 and perpendicular to the plane of the paper is
G = Fd,                                     (II)
where d is the lever-arm.
PROBLEMS.
1.   Prove that the moment of a force about an axis equals the moment of its component which lies in a plane perpendicular to the axis.
2.   Prove that the sum of the moments of the forces of a couple about any axis perpendicular to the plane of the couple is constant and equals the torque of the couple.
48.  ^Degrees of Freedom of a Rigid Body. — A rigid body may have a motion of translation along each of the axes of a rectangular system of coordinates and at the same time it can have a motion of rotation about each of these axes. Therefore a rigid body has six degrees of freedom, three of translation and three of rotation.    When one point in it is constrained to move in a plane the number of degrees of freedom is reduced to five.    When the point is constrained to move in a straight line the number becomes four.    When the point is fixed the body has only the three degrees of freedom of rotation.    If two points are fixed the body can only rotate about the line joining the two points.    Therefore its freedom is reduced to one degree.   When a third point, which is not in the line determined by the other two, is fixed the body cannot move at all, that is, it has no freedom of motion.
49.   The Law of Action and Reaction.—The law from which the conditions of equilibrium of a particle were obtained is a```